Wrapping Up Mirrors

We finished our investigation of mirrors today (the next assessment and subsequent reassessments notwithstanding); it was a 360 degree look at concave mirrors (ba dum ching!).

Students had used ray tracing to determine image locations for a variety of object locations - concave mirrors produce the biggest variety of image types and locations that we've seen, so each one was different in an important way.  They also collected data in the lab using the Pasco mirrors, light source, and screens, and tried to put all of this together into a nice predictive model for these images.

Things have been getting more difficult since we left plane mirrors (which were nice and predictable, with their upright, same size, same distance, virtual images).  Convex mirror images were always upright, virtual, and diminished, but the exact location and size have required tracing to determine.

Concave mirrors are even more complex, so after the data was taken, I set the groups off on five tasks, after which they reported back to the class with whiteboards, etc.:

• How do adjustable flashlights work?  What's being adjusted, and how does it make the beam go from narrow to parallel to wide?

Here they determined that the bulb position moving from inside the focus to the focus to outside of it was the key.

•  What does the image location vs. object location graph look like, and what does the magnification vs. object location graph look like?  This'll allow us to see the complicated trend and predict a bit.  We actually divided both of the image and object distances by the focal length so that we could use both the ray tracing data and the lab data together on the same graph.

Here they had to think about special-case points (like an object right on the mirror), recognize asymptotes (both vertical - for holes in the function - and horizontal - for long-term behavior), and draw the strangest fit line yet!

• Does this equation that I pulled out of thin air work for the images?

Here they saw that this equation does indeed work, assuming that your signs for all distances are correct.  It even works for convex mirrors (where the focal length is negative because the focus is behind the mirror) and for plane mirrors (if you take the focal length to be infinite, you get the equal-but-negative relationship that we've played with all term)!

Even better, this is exactly the kind of algebraic model that we thought that the graph suggested - the vertical asymptote and the flipped left and right sides strongly implied an inverse function, though it is offset, so it wasn't just a simple inverse function.  We had the right family, but it's a pretty tricky one to get just from the graph.

•  How about these two different definitions of magnification?

Some similar triangles saved the day here!

DIY Ballistic Pendulum

I was a bit tired of the old 1970's tiny teal ballistic pendulum that I had.  It works OK, but it's not terribly dramatic and it's quite small.

 Image: chinalabsupplies.com

Instead, I took a mailing tube from a cart track, filled it with a pocket of heavy fabric that I clipped around the edges, and mounted it on four strings in front of an air rocket launcher.  It took a couple of hours to do, including figuring out that I needed to mount it from the ceiling rack instead of a smaller setup that I originally had.

It will easily take the Super cap, and it's quite dramatic.  As I built it, the mass was roughly 12 times the rocket's mass, making for a pretty good ballistic pendulum.  Video here!

Momentum Transfer

Today we figured out how to incorporate gains and losses of momentum (impulse) into our model of momentum. First... let's predict how the Force vs. time and velocity vs. time graphs of a cart will look when it hits a hard bumper or a springy bumper (cue demo of cart doing this, always at the same initial v).

The whiteboards were awesome:

We went through to see what we had agreed upon and not (the "open questions"):

I was stoked about the great observations that so many of them made: the hard bumper made the cart come back more slowly, the elastic bumper was in contact longer, the velocity's sign changed (they all missed the sign of the force, but it was obvious what had happened when we saw the data from the force sensor!), the "hilly" nature of the force graph, the fact that the hard bumper exerted a higher peak force, etc.  There was even an awesome justification from one of the proponents of the "hill" (as opposed to the jump/spike in F) graphs that talked about how even the hard bumper gives a little bit (invoking a molecular "springiness" model), and so the force starts out small.  Literally - close to tears there.

You know the rest of the story - they saw that the elastic bumper transferred more momentum, even though it exerted a smaller peak force, reasoned that time was a factor, too, guessed that the area under the F vs. t graph might mean something, saw that it was just about exactly equal to the change in p (finally, some lab results worked super-well the first time :), and built our model for the change in momentum:

Or, for constant forces:

The best part was the appreciation of the analogy between work and impulse, which I think was more than a little due to the IFF diagrams - adding the initial momentum from the initial diagram to the area under the Fnet vs. t graph to get the momentum from the final diagram just makes it click for a ton of kids. We applied it to a sliding box, saw that it was easier than forces/kinematics, and talked for a free minute at the end of class about how physics used to be taught.  I described it to them (I needed to, since more than one said that she "couldn't imagine how else you could learn physics") as "throwing a bunch of equations at you and hoping that the concepts emerge from that and stick."  It worked for me, but not for most.  Reformed methods don't work for everybody either, but it's a huge increase in the proportion that really build an accurate mental model, without a doubt.

Taking Reflection Models for a Spin

In Physics this month we've been modeling reflection of light and the perception of images.  While we've applied that to both plane and curved mirrors, the principles are the same. A recent assessment, however, showed some confusion about the difference between specular (mirror-like) and diffuse (scattering) reflection - the reason that plywood is a terrible isn't that it's not reflective, but rather that its rough (at the micro-level) surface sends light going in many different directions. 

The other thing that usually takes a bit to really congeal for students is that the only way that we see things is for light to come from them (emission or reflection) and into our eyes.  There's a strong desire to believe that you could see a laser beam going by. Even though students have experience with laser pointers, they segment that experience and instead believe what they see on TV. :)

To attack both of these concepts, I had groups whiteboard some ray diagrams today to predict what they would see when they looked into a mirror on a wooden post.  They were set up at an angle to the mirror, and the lights would be turned off (and the blackout shades closed).  The only light would come from a flashlight that they would hold from their position that was not directly in front of the mirror:

I wanted them to tell me what they'd see in the mirror and what they'd see when they looked at the wood in this scenario.

They went to work whiteboarding, and more or less the only things that I'd say to them were: "does that follow the law of reflection?" and "how is it that you see things?"  The whole process of getting students to see for themselves when something that they've done contradicts a model that they know and agree with is a very difficult one.

Some of their boards:

Scientists and mathematicians really value consistency - every piece of evidence and analysis must tell the same story in order for the pieces to fit together comfortably, but students are extraordinarily willing to let contradictions sit there, frequently not being aware of them because they're not looking for connections and thinking about the implications of what they're seeing or producing.

After a bit, all of the groups had ray diagrams that fit the models, and which told them that we'd see the wood, but not the mirror.  There were just a few holdouts - students that couldn't quite believe that we could look at a mirror with a light shining on it and see just black.  Hopefully having enough time to really commit to their prediction made the evidence to the contrary enough to break the misconception, though that's a tricky process.

The black fabric below served to frame the scene a bit but had another purpose.  We get to contrast the fabric's properties with the wood's - both are rough, but the fabric absorbs a lot of light, while the wood's pretty reflective.  Even better, contrasting the black fabric (which absorbs lots of light, but isn't perfectly absorptive) with the black electrical tape securing it to the pole was another example of specular and diffuse reflection.  The tape also absorbs a lot, but its surface is much smoother than the fabric, so the light that's reflected went over to the other side of the room, meaning that the tape (like the mirror) appeared much darker than the fabric.

Modeling Plane Mirror Images

Last post, students modeled both reflection (using tennis balls) and the length necessary for a mirror to be "full length."  We've done quite a bit of ray tracing in our work with shadows, so I had them try to prove, using their reflection model (the Law of Reflection) and ray tracing, that the necessary length for a FLM was indeed half of the subject's height.

This was tricky for them - I had several students successfully find an example ray diagram that seemed to fit the model, but an example isn't a general proof.  Most of those students did some guessing-and-checking (as did pretty much all of the stragglers that did it in groups on whiteboards at the beginning of class.  I wasn't altogether happy with how this went as a HW exercise.  Some more scaffolding might be needed, but mostly there wasn't enough perseverance by most.

As an aside, the large majority did indeed bring in something written down, but there was a significant fraction that brought in something like these:


So folks really didn't know whether they had "done it right" or not.  This is an overarching critical thinking skills problem, not a physics problem.  It's one that we rarely address anywhere, though.  The idea here is that we have two paradigms controlling the situation:

• The angles of the incoming and reflected rays (preferably measured from the normal to the surface, but either will work at the moment) are equal
• We see things when light goes into our eyes after bouncing off of the object.  To see something in a mirror, the light bounces off of the object, then the mirror, then our eyes.

The first drawing there doesn't involve the eyes at all - the head and feet are involved, but there's a lack of understanding of the mechanism of seeing and/or of what the heck what we're drawing means.  The second drawing has massively unequal angles for each ray, and so is impossible.

This is where students need to build up the amount of fluency with the mental models that we use to see that these are ridiculous on their face.  It's like we've said "OMG, she tweeted me this 6 page sob story it wow soooo boring," or "he won the game with a touchdown in extra innings." They have mental models of those situations that show these statements, without effort or analysis, to be absurd.  The same isn't true for most of their 'academic' models.  The same things true for looking at a force diagram and saying "oh, that's obviously unbalanced, but it's supposed to be moving with constant velocity, so something's wrong."  Part of it is our job to pay attention to that, require it, talk about it in class, etc. and to connect what we study to the real world, so that students really do think that "real life" and "school" aren't disjoint sets.

So... folks eventually (and mostly by trial and error) got to here:

At this point, we get to talk about example vs. proof and how to make an abstract argument here about the relationship between the length of the mirror and the height of the person.  This actually hinges upon something that the non-guessers figured out, and in connects the "real world" back to the light discussion.  Just have two folks stand and bounce a ball to each other.  They bounce it off of the point on the floor halfway between them.  "Of course!" they say.  It's great to recognize it after the fact, but that model's just not in there firmly enough if they don't see it the first time. This allows us to really easily find the places on the mirror where those rays reflect, and to make a congruent triangles argument to show that the model that we came up with experimentally follows naturally from the law of reflection.

Images

OK, all of this guessing and checking is pain, and the midpoint trick only works if the observer and object are equidistant from the mirror.  We need a better way.  We did the classic "vertical piece of glass in a dark room with backlight mirror" trick - prop to glass on a sheet of paper, put a little lab weight in front of it, and maneuver an identical weight so that it always aligns with that image, when viewed from any angle.  Trace the object, image, and mirror a few times, and you can start to build a model to predict the size and location of an image.

The biggest idea that I'm pushing this year (so much so that I'm not having converging lenses, plane mirrors, etc. be separate standards) is the the location of the image is the point from which the light rays appear to diverge.  Everybody that sees the image points to it, and they only agree on one point.  It's a big picture that can get lost in all of the ray tracing rigamarole.

Each student's sheet (have each kid do one, it's quick) looks something like this:

The model follows pretty quickly at this point: the images are the same (perpendicular) distance behind the mirror that the objects are in front of it, and they're the same size as the objects. After making sure that they can do this for mirrors not aligned parallel to the paper's edges, we try to connect this again to the mechanism of seeing and the law of reflection.

First: have them pick one of the images and draw a couple of observers (these are supposed to be little eyes).  now, if we saw the image, and thought that it was there, what direction did the light rays have to come from?

 OK, good enough.  But... where did they really come from?

Hey - the law of reflection was upheld - automatically!  Look ma, no protractor!  Images are useful, quick, and easy.  Three great reasons for us to use them!

Let's bring it back to our ray tracing for the FLM.  It's sooooo much easier to just draw the image of the person (apologies, Randall) and work backwards to figure out where the rays came from, where they hit the mirror, etc.  This works great for figuring out the field of view of anyone, etc.  

The trickiest bit for the students is to see that this is exactly the same thing that we'll be doing with convex mirrors... and with concave mirrors... and with diverging lenses... and with converging lenses...  and with Coke glasses in restaurants... and with spearfishing... and with quarters in coffee cups.  Big picture, folks - keep their eyes on the prize.

Modeling Reflection and the Full-length Mirror

Having done quite well as a whole on our shadows, eclipses, and phases assessment before spring break, the physics classes moved on to another light-related topic: reflection.

First task:
     "Here's a ball.  Use it to construct a model of how light changes direction when it reflects off of a surface."

There was some very interesting work here, once everyone stopped considering the "boring case" (dropping the ball vertically onto a horizontal surface).  The biggest challenge was (as is frequently the case) experimental design.  There was lots of ball bouncing in vertical planes to begin with.  This is murderously hard to measure (looking at the angles before an after the bounces), and does that whole parabola thing that we don't really see light do!

Once everyone (literally) got rolling, things went quickly, and the law of reflection was quickly discovered.

We did a quickie whiteboard meeting all together - when everyone quickly came to the same conclusion, presenting it to each other is more than a little repetitive - where we came up with a single conclusion for the class.  I also mentioned that we might want to look at the angle between the normal and the ray, rather than the angle between the surface and the ray, because we'll deal with curved surfaces later, making the surface angle a little harder to measure, and because of this other thing that... well, you'll just need it later, so maybe get used to using this angle instead of the other.

As groups finished early, I put them on the hunt of limitations in our model: what does light do that the ball doesn't? ...what does the ball do that light doesn't?  Just like every other model, this one's an approximation - it gets some things quite well, but you need to be really aware of the things that it doesn't represent/predict well.

It was a great list:

• Light doesn't slow down when it reflects like the ball does
• Light can sometimes be absorbed by the surface, unlike the ball
• Light isn't usually noticeably affected by gravity (good little chat about black holes, gravitational lensing, etc.), unlike the ball
• Light isn't subject to friction/spin effects

We moved on to a challenge/lab next:

     "How long does a mirror need to be in order to be a full-length mirror?"

This was a great opportunity to define the problem, narrowing it down to make it doable, but keeping it broad enough to be useful, considering which assumptions were reasonable, etc.  It was also a chance to talk about doing a pilot experiment.  We had narrowed the task down to considering two variables: the person's height and his/her distance from the mirror.

 

At this point, we could go all crazy taking tons of data and then try to model it.  Instead, we discussed how, in real research, there's a cost to taking data which can be quite high.  For us, the cost is time.  Everybody remembered how much data they took when trying to determine the effect of amplitude on oscillator period, and we talked about how we could've saved that time by just taking a few data points that were well spread-out over the range of possible data, noticed that there wasn't an effect, and moved on.  The kids went off in groups, armed with small mirrors (they had to determine how to use these to determine how long the mirror would be for any given situation) and tasked with doing pilot studies on these two variables.  Some groups came to conclusions, and will refine their methods and take more comprehensive data on the variable(s) that matter, and other groups aren't quite there yet - we'll finish up next time.